| L(s) = 1 | − 2-s + 1.94·3-s + 4-s + 0.269·5-s − 1.94·6-s − 3.42·7-s − 8-s + 0.791·9-s − 0.269·10-s + 11-s + 1.94·12-s − 6.63·13-s + 3.42·14-s + 0.523·15-s + 16-s − 5.14·17-s − 0.791·18-s + 1.98·19-s + 0.269·20-s − 6.65·21-s − 22-s + 8.63·23-s − 1.94·24-s − 4.92·25-s + 6.63·26-s − 4.30·27-s − 3.42·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.12·3-s + 0.5·4-s + 0.120·5-s − 0.794·6-s − 1.29·7-s − 0.353·8-s + 0.263·9-s − 0.0850·10-s + 0.301·11-s + 0.562·12-s − 1.84·13-s + 0.914·14-s + 0.135·15-s + 0.250·16-s − 1.24·17-s − 0.186·18-s + 0.456·19-s + 0.0601·20-s − 1.45·21-s − 0.213·22-s + 1.80·23-s − 0.397·24-s − 0.985·25-s + 1.30·26-s − 0.827·27-s − 0.646·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.363631789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.363631789\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 - 1.94T + 3T^{2} \) |
| 5 | \( 1 - 0.269T + 5T^{2} \) |
| 7 | \( 1 + 3.42T + 7T^{2} \) |
| 13 | \( 1 + 6.63T + 13T^{2} \) |
| 17 | \( 1 + 5.14T + 17T^{2} \) |
| 19 | \( 1 - 1.98T + 19T^{2} \) |
| 23 | \( 1 - 8.63T + 23T^{2} \) |
| 29 | \( 1 - 1.68T + 29T^{2} \) |
| 31 | \( 1 - 4.13T + 31T^{2} \) |
| 37 | \( 1 + 1.79T + 37T^{2} \) |
| 41 | \( 1 - 8.13T + 41T^{2} \) |
| 43 | \( 1 - 9.37T + 43T^{2} \) |
| 47 | \( 1 - 6.44T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 3.06T + 61T^{2} \) |
| 67 | \( 1 + 2.93T + 67T^{2} \) |
| 71 | \( 1 + 5.44T + 71T^{2} \) |
| 73 | \( 1 + 5.55T + 73T^{2} \) |
| 79 | \( 1 - 5.78T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 0.575T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.658731780781389746673872526108, −7.50972402055690954045941500440, −7.24749437163867288196827623897, −6.46108055421176671133337334908, −5.55860353370179665677309036753, −4.46117993730907693325844884520, −3.50717073290991807155485220595, −2.53918724493501663573429304385, −2.40808129108931027382024586586, −0.65216434477047520462866048820,
0.65216434477047520462866048820, 2.40808129108931027382024586586, 2.53918724493501663573429304385, 3.50717073290991807155485220595, 4.46117993730907693325844884520, 5.55860353370179665677309036753, 6.46108055421176671133337334908, 7.24749437163867288196827623897, 7.50972402055690954045941500440, 8.658731780781389746673872526108
